A235121 - OEIS

%I #28 Sep 14 2024 13:22:27

%S 1,2,3,4,3,4,5,6,5,6,7,8,7,8,9,10,11,9,10,11,9,10,11,12,13,14,17,12,

%T 13,14,17,12,13,14,17,15,22,31,16,19,12,13,14,17,18,23,26,41,16,19,20,

%U 21,29,34,59,20,21,29,34,59,15,22,31,18,23,26,41,24,37,38,67,25,33,62,127

%N Irregular triangle read by rows: row n contains in increasing order the Matula numbers of the rooted trees that are isomorphic as trees to the rooted tree with Matula number n (n>=1).

%C Number of entries in row n is A235122(n).

%H Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.

%H Emerci Deutsch, <a href="http://dx.doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322.

%H F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.

%H I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.

%H I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.

%H D. Matula, <a href="http://www.jstor.org/stable/2027327?seq=30">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.

%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%F In order to construct the set A of numbers corresponding to row n, we start with A = {n} and we keep adjoining to A the numbers (x/p)' p", where x is an element of A, p is a prime factor of x, r' denotes the r-th prime and r" denotes the order of the prime r (i.e. r = r"-th prime). We do this until A becomes closed under the described operation. The Maple program (due to Edwin Clark) is based on this construction.

%e Row 11 is 9,10,11. Indeed the rooted tree with Matula number 11 is the path tree P[5] = ABCDE rooted at A; if rooted at B or D, then the Matula number is 10 and if rooted at C, then the Matula number is 9.

%e The triangle starts:

%e   1;

%e   2;

%e   3,4;

%e   3,4;

%e   5,6;

%e   5,6;

%e   7,8;

%e   ...

%p with(numtheory): f := proc (m) local x, p, S: S := NULL: x := factorset(m): for p in x do S := S, ithprime(m/p)*pi(p) end do: S end proc: M := proc (m) local A, B: A := {m}: do B := A: A := `union`(map(f, A), A): if B = A then return A end if end do end proc: for j to 20 do M(j) end do; # yields sequence in triangular form; from _W. Edwin Clark_

%t f[m_] := Prime[m/#]*PrimePi[#]& /@ FactorInteger[m][[All, 1]];

%t M[m_] := Module[{A, B}, A = {m}; While[True, B = A; A = Union[Map[f, A] // Flatten, A] // DeleteCases[#, 0]&; If[B == A, Return[A]]]];

%t Table[M[j], {j, 1, 25}] // Flatten (* _Jean-François Alcover_, Sep 14 2024, after _W. Edwin Clark_ *)

%Y Cf. A235122.

%Y Cf. A257539 (the first column).

%K nonn,tabf

%O 1,2

%A _Emeric Deutsch_, Jan 18 2014